 So one important idea is that we can't choose part of a term to multiply. So if our binomial terms have powers or additional factors, we have to include them. For example, let's try to find the x squared y to the fifth term in the expansion. So to get an x squared term, we'll have to choose a 3x term twice, but this means we'll have to choose the other term for y squared, 10 minus 2, 8 times. And so our term will be 10 choose 2, 3x twice, 4y squared, 8 times. Now, before we proceed, it's worth noting that there's a whole bunch of numbers here, but we're not going to worry about those immediately because we're looking for a particular term. And if we expand our powers, we see that we get an x squared y to the 16th, and this won't be an x squared y to the fifth term. And you should convince yourself that there's no other way we can even get an x squared, and so there won't be an x squared y to the fifth term. Well, what about an x squared term, and we don't know what the y is? So here we mean that the term has x squared and possibly other factors that aren't x. So again, we'd get x squared by choosing 3x twice and 4y squared 8 times. So of our 10 factors, 2 of them will be 3x's and 8 for y squared's. And we'll compute, well, at least we'll make sure our variable terms work out. We're not going to bother multiplying out the numbers. What if we wanted the x cubed term in the expansion of x plus 1 divided by x raised to power 7? So we might start out by saying, well, we'd get an x cubed term by choosing x 3 times, but then we'd have to choose 1 divided by x 4 times, and our actual term would be 7 choose 3 x cubed 1 divided by x to the fourth. Again, ignoring our numerical values, the variable parts of this expression will simplify down to 1 divided by x, and this is not x cubed. Now what's different here is that we could incorporate that other factor and possibly get an x cubed. So let's think about this. To get an x cubed term, we need to choose x k times, don't know how many that is, and 1 divided by x the remaining 7 minus k times. And what we'd want is that our variables, x to the k times 1 divided by x to the power 7k, has to be x cubed. And we can do a little bit of algebraic simplification here. We can do a little more than that. One more. And what we want is our exponents to be the same. So we want 2k minus 7 to be equal to 3. We can solve. And that tells us to get an x cubed term, we need to choose our x term 5 times and 1 divided by x 2 times. And so our actual term will be 7 choose 5, x chosen 5 times, and 1 divided by x chosen 2 times. And we can expand and simplify, which gets us our x cubed term. So let's try to find our y cubed term in our expansion x squared y plus x divided by y squared to power 8. And maybe we can find it. Maybe we can't. Let's see. So we have to choose x squared y some number of times k times and x divided by y squared the remaining 8 minus k times. So we'd want x squared y to power k times x divided by y squared to power 8 minus k to be something y cubed. And we'll do a little bit of algebraic simplification. And we don't actually care about the power on x because this y cubed term could include factors of x that we don't really keep track of, but we do want to make sure that the exponent on y is equal to 3. And so we'd want 3k minus 16 to be equal to 3. Solving. And so we want to choose x squared y 19 third times. And that's a problem. We can't choose it a fractional number of times. And so that says there's no y cubed term in the expansion. And again, it's because this equation doesn't have a whole number solution so there can be no y cubed term. What about a y to the 11th? Well, again, we want to choose x squared y k times and the other term 8 minus k times. And we want to make sure that when we do that we get a y to the 11th plus possibly some other things. Do a little algebra. And we want 3k minus 16 to equal 11. And solving. And this time we have a whole number solution so all we need to do is pick x squared y 9 times and there's a problem because there's only 8 factors to choose from. And since this would require more factors of x squared y than we have there is no y to the 11th term. Well, how about a y to the 5th term? So we'll go through the process and we want 3k minus 16 to equal 5. And we can choose x squared y 7 times and the other factor the remaining 8 minus 7 one time. And so we can get a y to the 5th term by choosing x squared y 7 times and x divided by y squared once and that term will be 8 choose 7 x squared y to the 7th x divided by y squared to the 1th and we compute. To get our y to the 5th term.